Many religions claim we came to this earth on a mission, everyone has a different mission, a specific reason to be here.

In Buddhism I couldn't find anything like that, the "mission" would be the same for every one: Become an Arahant and scape samsara. There is nothing we should achieve here, except from that. Is that a fair statement?

Does anyone know a sutta, speech or book related to Buddhism that states otherwise? That we do have a specific mission on this planet?

In Buddhism there are different levels of motivation, on the "lower" levels of motivation are the range of goals to escape suffering for oneself only.

In "higher" scopes of motivation there is the goal to not only escape suffering, but to help others escape as well.

The "highest" scope would be a goal to help all beings escape suffering (Bodhichitta)

I put the words lower, higher, and highest in parenthesis as they are what I believe to be part of a false critique the mahayana/vajrayana holds of the other schools. In reality the other schools are a prereq to even attempting Mahayana practices in most cases. I don't like or agree with the critique but it helps to explain.

The Bodhisattva vows and the goal of the higher schools could be considered a mission. But is not a requirement in Buddhism.

For analogy, in war time, the ultimate mission of the military is to win the war. But to achieve that mission, many sub-missions would also need to be completed successfully: winning battles on various fronts: intelligence, logistics, politic, economic, etc. Similarly for Buddhism, to successfully defeat Mara's army and put an end to Samsara, there're many "sub-missions" to be done. A lot of these like observing the precepts, practicing loving-kindness, generosity, etc. are also common missions across religions. Ven. Bodhi in his "In the Buddha's Words" did a good job in organizing the Buddha's teaching into systematic and gradual Dhamma "missions": Happiness visible in this present life, the way to a fortunate rebirth, mastering the mind and the Path to Liberation, attaining the Light of Wisdom, and ultimately, Final Liberation.

You wrote, "everyone has a different mission, a specific reason to be here" and maybe this story is a Buddhist version of that: i.e. these specific women had a personal feud, which continued through rebirths, until eventually they learned to stop feuding.

You could phrase that as saying that their "specific reason to be here" was to learn that "hatred could only cause more hatred, and that it could only cease through friendship, understanding and goodwill".

At the end of the discourse, the ogress was established in Sotapatti Fruition and the long-standing feud came to an end.

In the Dhp. 5 story, the actions of the husband who took a second wife causing much hurt to his first wife seem like they should net him some bad karma, but the story does not mention it - which leads me to think in ancient India it was the done thing - and thus karma neutral. In today's society it would have ended in a bitterly contested divorce - with the husband and wife being reborn as animals to continue fighting.
– BuddhoSep 28 '15 at 14:45

Simply put, your mission is to be absolutely happy, happiness that does not depend on circumstances but comes forth through your own Buddha nature manifesting. At the same time, enabling others to do the same. In this Saha world which is full of suffering and negativity, in order to do that one needs to "demonstrate" to others how.

This is where our desires and social roles come into play. To create value as a family member, doctor, teacher (pick your profession), to be able to create value in other ways in society centered on Buddhist practice, etc is the "role" one plays to display the immense power of Buddhism and lead others to absolute happiness. Buddhism and daily life are one and the same.

To say that our mission as Buddhists is to escape Samsara (this world) or in other words to escape from the cycle of birth and death (wrongly interpreted as nirvana) is to be selfish. That is not the way a Buddha or a Bodhisattva thinks.

Shakyamuni Buddha says in the Expedient Means chapter of the Lotus Sutra -
"At all times, I think to myself: how can I cause living beings to gain entry into the unsurpassed way and quickly acquire the body of a Buddha?"

Note that Lord Buddha says that everyone can become a Buddha, and he says that at all times he exerts himself to enable others to become like him.

In short, to enable ourselves to reveal our Buddha nature and helping others do the same is our true mission as Buddhists.